An Analysis of Probabilistic Forwarding of Coded Packets on Random Geometric Graphs

We consider the problem of energy-efficient broadcasting on dense ad-hoc networks. Ad-hoc networks are generally modeled using random geometric graphs (RGGs). Here, nodes are deployed uniformly in a square area around the origin, and any two nodes which are within Euclidean distance of $1$ are assumed to be able to receive each other's broadcast. A source node at the origin encodes $k$ data packets of information into $n\ (>k)$ coded packets and transmits them to all its one-hop neighbors. The encoding is such that, any node that receives at least $k$ out of the $n$ coded packets can retrieve the original $k$ data packets. Every other node in the network follows a probabilistic forwarding protocol; upon reception of a previously unreceived packet, the node forwards it with probability $p$ and does nothing with probability $1-p$. We are interested in the minimum forwarding probability which ensures that a large fraction of nodes can decode the information from the source. We deem this a \emph{near-broadcast}. The performance metric of interest is the expected total number of transmissions at this minimum forwarding probability, where the expectation is over both the forwarding protocol as well as the realization of the RGG. In comparison to probabilistic forwarding with no coding, our treatment of the problem indicates that, with a judicious choice of $n$, it is possible to reduce the expected total number of transmissions while ensuring a near-broadcast.